The fractional logarithmic Schrödinger operator: properties and functional spaces

In this note, we deal with the fractional logarithmic Schrödinger operator \((I+(-\Delta )^s)^{\log }\) ${(I+{(-\mathrm{\Delta })}^s)}^{log}$ and the corresponding energy spaces for variational study. The fractional (relativistic) logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, \(\log (1+|\xi |^{2s})\) $log(1+|\xi {|}^{2s})$ , \(s>0\) $s>0$ . We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of \({\mathbb {R}}^N.\) ${\mathbb{R}}^N.$ We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity.